the Wilcoxon Rank Sum Test and the Wilcoxon Signed-Rank Test are important non-parametric methods used when the assumptions of parametric tests cannot be justified.
These tests are particularly useful in medical and biological research where data may not be normally distributed, sample sizes are small, or the data are ordinal rather than interval or ratio scale.
Wilcoxon Rank Sum Test (for two independent samples)
The Wilcoxon Rank Sum Test, also known as the Mann-Whitney U test, is used in biostatistics to compare two independent groups when the data are not normally distributed.
This might include comparing the effect of two different treatments on a continuous outcome that is skewed or ordinal.
Steps to Perform the Wilcoxon Rank Sum Test:
Combine and Rank the Data: Take all observations from both groups and rank them from the smallest to largest. If there are ties (equal values), assign the average of the ranks that would have been assigned had the values been slightly different.
Calculate Sum of Ranks: For each group, calculate the total of the ranks. Let these sums be R1 for sample 1 and 𝑅2 for sample 2.
3. Compute Test Statistic: The test statistic can be computed as 𝑈1 or 𝑈2, using the formulas:
Here, 𝑛1 and 𝑛2 are the sample sizes of the two groups. Usually, the test statistic 𝑈 is the smaller of 𝑈1 and 𝑈2.
4. Determine the P-value: Refer to statistical tables or use software to determine the p-value for 𝑈. For larger sample sizes, a normal approximation can be employed.
5. Interpret Results: If the p-value is less than the chosen significance level (often 0.05), reject the null hypothesis that the two populations have the same distribution.
Wilcoxon Signed-Rank Test (for paired data)
In biostatistics, the Wilcoxon Signed-Rank Test is used for paired data, such as before-and-after measurements in a treatment study, or matched pairs in case-control studies.
Steps to Perform the Wilcoxon Signed-Rank Test:
Compute Differences: Calculate the difference 𝐷𝑖 = 𝑋𝑖 − 𝑌𝑖 for each pair of observations 𝑋𝑖 and 𝑌𝑖.
Rank the Absolute Differences: Ignore differences of zero and rank the absolute values ∣𝐷𝑖∣.
Assign Signs: Attach the sign of each difference 𝐷𝑖 to its corresponding rank.
Sum the Positive and Negative Ranks: Let 𝑊+ be the sum of the ranks for positive differences and 𝑊− the sum for negative differences.
Compute the Test Statistic: Typically, the test statistic 𝑊 is the smaller of 𝑊+ and W−.
Determine the P-value: Use statistical tables or software to find the p-value based on 𝑊 and the number of pairs.
Interpret Results: If the p-value is below the significance level, reject the null hypothesis that the median of the differences is zero, indicating a significant effect or difference between the paired measurements.
Wilcoxon Signed-Rank Test (for a single set of data)
When you have a single sample and want to compare it against a hypothetical median, the process is similar to the paired test but focuses on the difference from a hypothetical value.
Steps to Perform the Test:
Compute Differences: For each observation 𝑋𝑖, calculate 𝐷𝑖 = 𝑋𝑖 − 𝑚, where 𝑚 is the hypothetical median.
Follow the Same Steps 2-7: Continue as if these were paired differences from a null hypothesis median of zero.
In biostatistics, these tests are invaluable because they allow researchers to make robust inferences about their data without the strict assumptions of normality or equal variances required by parametric tests. They are widely used in areas like clinical trials, ecological studies, and other biomedical research fields.